Complexity of Validity of Backround Propositional Dependence Logics - PowerPoint PPT Presentation

Complexity of Validity of Propositional Dependence Logics Jonni Virtema Complexity of Validity of Backround Propositional Dependence Logics Propositional logics Team semantics Modal dependence Jonni Virtema logic Complexity Japan Advanced Institute of Science and Technology, Japan University of Tampere, Finland jonni.virtema@uta.fi GandALF 2014 10th of September, 2014

Complexity of Motivation and history Validity of Propositional Dependence Logics Logical modelling of uncertainty, imperfect information and functional Jonni Virtema dependence in the framework of propositional (modal) logic. Backround Propositional The ideas are transfered from first-order dependence logic (and logics independence-friendly logic) to propositional (modal) logic. Team semantics Modal dependence Historical development: logic ◮ Branching quantifiers by Henkin 1959. Complexity ◮ Independence-friendly logic by Hintikka and Sandu 1989. ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ IF modal logic by Tulenheimo 2003. ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008.

Complexity of Syntax for propositional logics Validity of Propositional Dependence Logics Jonni Virtema Definition Backround Let Φ be a set of atomic propositions. The set of formulae for propositional logic Propositional PL (Φ) is generated by the following grammar logics Team semantics ϕ ::= p | ¬ p | ( ϕ ∨ ϕ ) | ( ϕ ∧ ϕ ) , Modal dependence logic where p ∈ Φ. Complexity The syntax for standard modal logic ML (Φ) extends the syntax for PL (Φ) by the grammar rules ϕ ::= ♦ ϕ | � ϕ. Note that formulas are assumed to be in negation normal form: negations may occur only in front of atomic formulas.

Complexity of Semantics for propositional logics Validity of Propositional Dependence Logics The semantics for PL (Φ) and ML (Φ) could be defined as usual, i.e., with Jonni Virtema assignments and pointed Kripke models, respectively. Backround In order to simplify the presentation, at this point, we consider propositional Propositional logics logic as a fragment of modal logic without modalities. Team semantics Modal dependence Definition logic Complexity Let Φ be a set of atomic propositions. A Kripke model K over Φ is a tuple K = ( W , R , V ) , where W is a nonempty set of worlds , R ⊆ W × W is a binary relation, and V is a valuation V : Φ → P ( W ). We will give team semantics for PL (Φ) and ML (Φ).

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema Backround Propositional logics Team semantics Modal dependence logic Complexity

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics Team semantics Modal dependence logic Complexity

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics 2. The standard semantics for modal logic is given with respect to pointed Team semantics models K , w . In team semantics the semantics is given for models and Modal dependence logic teams, i.e., with respect to pairs K , T , where T is a team of K . Complexity

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics 2. The standard semantics for modal logic is given with respect to pointed Team semantics models K , w . In team semantics the semantics is given for models and Modal dependence logic teams, i.e., with respect to pairs K , T , where T is a team of K . Complexity 3. Some possible interpretations for K , w and K , T :

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics 2. The standard semantics for modal logic is given with respect to pointed Team semantics models K , w . In team semantics the semantics is given for models and Modal dependence logic teams, i.e., with respect to pairs K , T , where T is a team of K . Complexity 3. Some possible interpretations for K , w and K , T : (a) K , w | = ϕ : The actual world is w and ϕ is true in w .

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics 2. The standard semantics for modal logic is given with respect to pointed Team semantics models K , w . In team semantics the semantics is given for models and Modal dependence logic teams, i.e., with respect to pairs K , T , where T is a team of K . Complexity 3. Some possible interpretations for K , w and K , T : (a) K , w | = ϕ : The actual world is w and ϕ is true in w . (b) K , T | = ϕ : The actual world is in T , but we do not know which one it is. The formula ϕ is true in the actual world.

Complexity of Team semantics? Validity of Propositional Dependence Logics Jonni Virtema 1. In this context a team is a set of possible worlds, i.e., if K = ( W , R , V ) is a Backround Propositional Kripke model then T ⊆ W is a team of K . logics 2. The standard semantics for modal logic is given with respect to pointed Team semantics models K , w . In team semantics the semantics is given for models and Modal dependence logic teams, i.e., with respect to pairs K , T , where T is a team of K . Complexity 3. Some possible interpretations for K , w and K , T : (a) K , w | = ϕ : The actual world is w and ϕ is true in w . (b) K , T | = ϕ : The actual world is in T , but we do not know which one it is. The formula ϕ is true in the actual world. (c) K , T | = ϕ : We consider sets of points as primitive. The formula ϕ describes properties of collections of points.

Complexity of Team semantics for modal logic Validity of Propositional Dependence Logics Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Propositional logics K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence K , w | = p ⇔ w ∈ V ( p ) . logic K , w | = ¬ p ⇔ w / ∈ V ( p ) . Complexity K , w | = ϕ ∧ ψ ⇔ K , w | = ϕ and K , w | = ψ. K , w | = ϕ ∨ ψ ⇔ K , w | = ϕ or K , w | = ψ. K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔

Complexity of Team semantics for modal logic Validity of Propositional Dependence Logics Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Propositional logics K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence K , T | = p ⇔ T ⊆ V ( p ) . logic K , T | = ¬ p ⇔ T ∩ V ( p ) = ∅ . Complexity K , T | = ϕ ∧ ψ ⇔ K , T | = ϕ and K , T | = ψ. K , w | = ϕ ∨ ψ ⇔ K , w | = ϕ or K , w | = ψ. K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔

Complexity of Team semantics for modal logic Validity of Propositional Dependence Logics Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Propositional logics K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence K , T | = p ⇔ T ⊆ V ( p ) . logic K , T | = ¬ p ⇔ T ∩ V ( p ) = ∅ . Complexity K , T | = ϕ ∧ ψ ⇔ K , T | = ϕ and K , T | = ψ. K , T | = ϕ ∨ ψ ⇔ K , T 1 | = ϕ and K , T 2 | = ψ for some T 1 ∪ T 2 = T . K , w ′ | = ϕ for every w ′ s.t. wRw ′ . K , w | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔

Complexity of Team semantics for modal logic Validity of Propositional Dependence Logics Jonni Virtema Definition Backround Kripke/Team semantics for ML is defined as follows. Remember that Propositional logics K = ( W , R , V ) is a normal Kripke model and T ⊆ W . Team semantics Modal dependence K , T | = p ⇔ T ⊆ V ( p ) . logic K , T | = ¬ p ⇔ T ∩ V ( p ) = ∅ . Complexity K , T | = ϕ ∧ ψ ⇔ K , T | = ϕ and K , T | = ψ. K , T | = ϕ ∨ ψ ⇔ K , T 1 | = ϕ and K , T 2 | = ψ for some T 1 ∪ T 2 = T . K , T ′ | = ϕ for T ′ := { w ′ | w ∈ T , wRw ′ } . K , T | = � ϕ ⇔ K , w ′ | = ϕ for some w ′ s.t. wRw ′ . K , w | = ♦ ϕ ⇔